If $V_{d,\delta }$ denotes the variety of irreducible plane curves of degree $d$ with exactly $\delta$ nodes as singularities, Diaz and Harris (1986) have conjectured that $\mathrm{Pic}\left(V_{d,\delta }\right)$ is a torsion group. In this note we study rational equivalence on some families of singular plane curves and we prove, in particular, that $\mathrm{Pic}\left(V_{d,1}\right)$ is a finite group, so that the conjecture holds for $\delta =1$. Actually the order of $\mathrm{Pic}\left(V_{d,1}\right)$ is $6(d-2)d^2-3d+1)$, the group being cyclic if $d$ is odd and the product of ${\mathbb{Z}}_2$ and a cyclic group of order $3(d-2)(d^2-3d+1)$ if $d$ is even.